Higher dimensional homological algebra

Higher dimensional homological algebra Peter Jørgensen Contents 1 Preface 3 2 Notation and Terminology 5 3 d-cluster tilting subcategories 6 4 Higher Auslander Reiten translations 10 5 d-abelian categories 15 1

6 Higher Auslander Reiten sequences 19 7 (d + 2)-angulated categories 21 A A class of examples by Vaso 26 A.i The d-abelian category of Φ............. 28 A.ii The (d + 2)-angulated category of Φ........ 33 2

1 Preface These are notes on higher dimensional homological algebra for the LMS CMI Research School New trends in representation theory The impact of cluster theory in representation theory, held at the University of Leicester 19 23 June 2017. The notes take the view that d-abelian and (d + 2)-angulated categories, as introduced by Jasso and Geiss Keller Oppermann, are the natural framework for higher dimensional homological algebra. The theory of d-cluster tilting subcategories and higher Auslander Reiten theory are developed as means to this end. Many important parts of the theory are not covered: The higher Auslander correspondence, d-exact categories, d-frobenius categories, relative d-cluster tilting subcategories, etc. Appendix A describes a class of examples by Vaso. We do not describe any of the other examples which are known: Higher Auslander, Nakayama, and preprojective algebras, Iyama s cone construction, etc. In particular, we do not explain their rich combinatorial structure, which is one of the motivations for higher homological algebra. 3

None of the material in these notes is due to me. I have tried to give due credit for all definitions and results, but would be grateful to learn if there are missing, incomplete, or wrong attributions. I thank the organisers of the LMS CMI Research School, Karin Baur and Sibylle Schroll, for the opportunity to give a course, and for the chance to present these notes. I thank Gustavo Jasso for providing feedback on a preliminary version, and for providing a number of exercises. 4

2 Notation and Terminology Symbol Meaning d A positive integer Φ A finite dimensional C-algebra mod(φ) The category of finite dimensional right Φ-modules mod(φ) mod(φ) modulo morphisms which factor through a projective mod(φ) mod(φ) modulo morphisms which factor through an injective mod(φ op ), mod(φ op ), mod(φ op ) The same for left Φ-modules F mod(φ) A d-cluster tilting subcategory F F modulo morphisms which factor through a projective F F modulo morphisms which factor through an injective D( ) Hom C (, C) ( ) Hom Φ (, Φ) if is a Φ-module Tr d The d th higher transpose τ d = D Tr d The d-auslander Reiten translation τ d = Tr d D The inverse d-auslander Reiten translation 5

3 d-cluster tilting subcategories Definition 3.1. Let A be a category, F A a full subcategory. Then F is called: (i) Generating if each a A permits an epimorphism f a with f F, (ii) Cogenerating if each a A permits a monomorphism a f with f F. Definition 3.2 (Enochs [1]). Let A be a category, F A a full subcategory. A morphism a α f in A with f F is called: An F -preenvelope of a if it has the extension property a α f f for each morphism a f with f F. An F -envelope of a if it is an F -preenvelope which is left minimal, that is, satisfies that each morphism f ϕ f with ϕα = α is an automorphism. 6

We say that F is preenveloping (resp. enveloping) in A if each a A has an F -preenvelope (resp. envelope). The dual notion of preenvelope is precover, and F is called functorially finite if it is preenveloping and precovering. Definition 3.3 (Iyama [3, def. 2.2], Jasso [5, def. 3.14]). Let A be an abelian or triangulated category. A d-cluster tilting subcategory F of A is a full subcategory which satisfies the following. (i) F = {a A Ext 1..d 1 (F, a) = 0} = {a A Ext 1..d 1 (a, F ) = 0}. (ii) F is functorially finite. (iii) If A is abelian then F is generating and cogenerating. A d-cluster tilting object f of A is an object f such that F = add(f) is a d-cluster tilting subcategory. Remark 3.4. If p A is projective, then 3.3(i) implies p F. Similarly, if i A is injective, then i F. Hence if A has enough projectives and injectives, then 3.3(i) implies 3.3(iii). 7

Definition 3.5. Let A be an abelian category, F a full subcategory. An augmented left F -resolution of a A is a sequence f 2 f 1 f 0 a 0 with f i F for each i, which becomes exact under A (f, ) for each f F. Then f 2 f 1 f 0 0 is called a left F -resolution of a. The resolutions are said to have length n if f j = 0 for j n + 1. (Augmented) right resolutions are defined dually. Proposition 3.6 (Iyama [3, prop. 2.2.2]). Let A be an abelian category, F mod(φ) a d-cluster tilting subcategory. Then each a A has (augmented) left and right F -resolutions of length d 1. Proof. We give a proof for left resolutions in the special case d = 2. There is an F -precover f ϕ a by Definition 3.3(ii), and ϕ is an epimorphism by Exercise 3.7. Hence there is a short exact sequence 8

0 k f sequence ϕ a 0. If f F then there is a long exact 0 Hom Φ ( f, k) Hom Φ ( f, f) ϕ Hom Φ ( f, a) (1) Ext 1 Φ( f, k) Ext 1 Φ( f, f). Here ϕ is surjective since ϕ is an F -precover and Ext 1 Φ( f, f) = 0 by Definition 3.3(i) whence Ext 1 Φ( f, k). This shows k F. It also shows that 0 Hom Φ ( f, k) Hom Φ ( f, f) Hom Φ ( f, a) 0 is exact for each f F whence 0 k f a 0 is an augmented left F -resolution of a. Exercise 3.7. Let A be an abelian category, F a full subcategory. (i) Show that if F is generating, then each F -precover is an epimorphism. (ii) Show that if F is cogenerating, then each F -preenvelope is a monomorphism. Exercise 3.8. Let R be a commutative ring, A an R-linear Hom-finite Krull Schmidt category, F a full subcategory closed under sums and summands. Show that if F has only finitely many isomorphism classes of indecomposable objects, then it is functorially finite. Exercise 3.9. Let A be an abelian category, F a full subcategory. Show that if F is precovering, then each a A has an augmented left F -resolution which can be obtained as follows: Set k 1 = a. Suppose k i has been defined. Pick an F -precover f j+1 k j and complete to a left exact sequence 0 k j+1 f j+1 k j. Exercise 3.10. Prove Proposition 3.6 in general. 9

4 Higher Auslander Reiten translations Definition 4.1 (Iyama [3, 1.4.1]). Let m mod(φ) have the augmented projective resolution p 2 p 1 p 0 m 0. The d th higher transpose of m is Tr d (m) = Coker(p d 1 p d ). The d-auslander Reiten translation of m and the inverse d-auslander Reiten translation of m are τ d (m) = D Tr d (m), τ d (m) = Tr d D(m). Proposition 4.2. There are functors mod(φ) Tr d mod(φ op ) and mod(φ) τ d τ d mod(φ). Lemma 4.3. Let F mod(φ) be a d-cluster tilting subcategory. Let f F have the augmented projective resolution p 2 p 1 p 0 f 0. 10

Then Tr d (f) has an augmented projective resolution which begins p 0 p 1 p d 1 p d Tr d (f) 0, (2) and τ d (f) has an augmented injective resolution which begins 0 τ d (f) D(p d ) D(p d 1) D(p 1 ) D(p 0 ). Proof. Each module p j = Hom Φ(p j, Φ) is projective. The last part of the augmented projective resolution, p d 1 p d Tr d (f) 0, is exact by the definition of Tr d (f). If 1 j d 1 then the homology of (2) at p j is Ext j Φ (f, Φ) by definition, and this Ext is 0 since f, Φ F. The augmented injective resolution is obtained by dualising the augmented projective resolution. Lemma 4.4. For m, p mod(φ) with p projective, there is a natural isomorphism Hom Φ ( m, D(p ) ) = DHomΦ (p, m). 11

Proof. We can compute as follows, using tensor-hom adjointness (a) and an evaluation morphism (b). Hom Φ ( m, D(p ) ) = HomΦ ( m, Hom C ( HomΦ (p, Φ), C )) (a) ( = HomC m Hom Φ (p, Φ), C ) Φ (b) ( = HomC HomΦ (p, m Φ), C ) Φ ( = HomC HomΦ (p, m), C ) = DHom Φ (p, m). Theorem 4.5 (Iyama [3, thm. 2.3]). Let F mod(φ) be a d- cluster tilting subcategory. The τ d and τ d map F to itself. Proof. We show the statement for τ d. Let f, f F and 1 j d 1 be given. Lemma 4.3 gives an injective resolution of τ d (f). Applying Hom Φ (f, ) gives a complex which computes the Ext j Φ( f, τ d (f) ). By Lemma 4.4, the complex is isomorphic to the following. DHom Φ (p d, f ) DHom Φ (p 0, f ) 0 12

The homology at DHom Φ (p j, f ) is DExt j Φ (f, f ) by definition, and this Ext is zero since f, f F. Hence Ext j Φ( f, τ d (f) ) = 0 whence τ d (f) F since f F and 1 j d 1 are arbitrary. Theorem 4.6 (Iyama [3, thm. 2.3]). Let F mod(φ) be a d- cluster tilting subcategory. There are quasi-inverse equivalences F τ d τ d F. Proof. Combine Proposition 4.2 and Theorem 4.5 to get the existence of the functors. It is a separate argument to show that they are quasi-inverse to each other. Definition 4.7 (Iyama Oppermann [4, def. 2.2]). The algebra Φ is called d-representation finite if gldim(φ) d and Φ has a d-cluster tilting object. Theorem 4.8 (Iyama). If Φ is d-representation finite, then mod(φ) has the unique d-cluster tilting subcategory F = add{ τ j d (i) i injective in mod(φ) and 0 j }. Exercise 4.9. Prove Proposition 4.2. Exercise 4.10. Show that a d-representation finite algebra has global dimension 0 or d. 13

Exercise 4.11. Let Φ be the path algebra of the following quiver with the indicated relation. Show that Φ is 2-representation finite. Exercise 4.12. Let Φ be the path algebra of the following quiver with the indicated relations. Show that Φ is 2-representation finite. 14

5 d-abelian categories Definition 5.1 (Jasso [5, defs. 2.2 and 2.4]). Let F be an additive category. (i) A diagram f d+1 f 2 f 1 is a d-kernel of a morphism f 1 f 0 if 0 f d+1 f 2 f 1 f 0 becomes an exact sequence under F ( f, ) for each f F. (ii) A diagram f d f d 1 f 0 is a d-cokernel of a morphism f d+1 f d if f d+1 f d f d 1 f 0 0 becomes an exact sequence under F (, f) for each f F. (iii) A d-exact sequence is a diagram f d+1 f d f d 1 f 2 f 1 f 0 such that f d+1 f 2 f 1 is a d-kernel of f 1 f 0 and f d f d 1 f 0 is a d-cokernel of f d+1 f d. 15

Remark 5.2. A d-exact sequence is often written 0 f d+1 f d f d 1 f 2 f 1 f 0 0. (3) This is motived by Exercise 5.7. Definition 5.3 (Jasso [5, def. 3.1]). An additive category F is called d-abelian if it satisfies the following: (A0) F has split idempotents. (A1) Each morphism in F has a d-kernel and a d-cokernel. (A2) If f d+1 f d is a monomorphism which has a d-cokernel f d f 0, then 0 f d+1 f d f 0 0 is a d-exact sequence. (A2 op ) The dual of (A2). Conditions (A2) and (A2 op ) can be replaced with: (A2 ) If f d+1 f d is a monomorphism, then there exists a d-exact sequence f d+1 f d f 0. 16

(A2 op ) The dual of (A2 ). Remark 5.4. The notions of 1-kernel, 1-cokernel, 1-exact sequence, and 1-abelian category coincide with the notions of kernel, cokernel, short exact sequence, and abelian category. Theorem 5.5 (Jasso [5, thm. 3.16]). If F is a d-cluster tilting subcategory of an abelian category A, then F is d-abelian. Remark 5.6. We will not prove Theorem 5.5, but will explain how to obtain d-kernels and d-cokernels in F : ϕ Let f 1 1 f0 be a morphism in F. Viewed as a morphism in A, is has a kernel k left F -resolution κ f 1. By Proposition 3.6 there is an augmented 0 f d+1 f 2 ε k 0. Then f d+1 f 2 κε f 1 is a d-kernel of ϕ 1. ϕ d+1 Let f d+1 f d be a morphism in F. Viewed as a morphism in A, ϕ is has a cokernel f d c. By Proposition 3.6 there is an augmented 17

right F -resolution 0 c γ f d 1 f 0 0. Then f d γϕ f d 1 f 0 is a d-cokernel of ϕ d+1. Exercise 5.7. (i) Let f d+1 f 2 f 1 be a d-kernel of f 1 f 0. Show that f d+1 f d is a monomorphism. (ii) Let f d f d 1 f 0 be a d-cokernel of f d+1 f d. Show that f 1 f 0 is an epimorphism. Exercise 5.8. (i) Show that 1-kernel means kernel. (ii) Show that 1-cokernel means cokernel. (iii) Show that 1-exact sequence means short exact sequence. (iv) Show that 1-abelian category means abelian category. Exercise 5.9. Show that the d-kernels and d-cokernels constructed in Remark 5.6 have the properties required by Definition 5.1. Exercise 5.10. (i) Show that kernels and cokernels (=1-kernels and 1-cokernels) are unique up to unique isomorphism. (ii) Give an example to show that if d 2, then d-kernels and d-cokernels are not unique up to isomorphism. 18

6 Higher Auslander Reiten sequences Definition 6.1. Let F be a category. ϕ (i) A morphism f 1 1 f0 is a split epimorphism if it has a right ϕ inverse, that is, if there is a morphism f 0 0 f1 such that ϕ 1 ϕ 0 = id f0. ϕ (ii) A morphism f 1 1 f0 is right almost split if it is not a split epimorphism and has the lifting property f f 1 ϕ 1 f 0 for each morphism f f 0 which is not a split epimorphism. ϕ (iii) A morphism f 1 1 f0 is right minimal if each morphism ϕ f 1 with ϕ 1 ϕ = ϕ 1 is an isomorphism. f 1 There are dual notions of split monomorphism, left almost split morphism, and left minimal morphism. The notion of d-auslander Reiten sequences is due to Iyama, see [3]. The following version of the definition is due to Jasso and Kvamme. 19

Definition 6.2 ([6, def. 5.3]). A d-auslander Reiten sequence in a d-abelian category F is a d-exact sequence ϕ d+1 0 f d+1 fd ϕ d ϕ d 1 ϕ 3 ϕ 2 ϕ 1 fd 1 f2 f1 f0 0 (4) such that ϕ d+1 if left almost split and left minimal, ϕ 1 is right almost split and right minimal, and ϕ d,..., ϕ 2 are in the radical of F. Theorem 6.3 ([3, thm. 3.3.1]). Let F mod(φ) be a d-cluster tilting subcategory. (i) For each non-projective indecomposable object f 0 F, there exists a d-auslander Reiten sequence (4) in F. (ii) For each non-injective indecomposable object f d+1 F, there exists a d-auslander Reiten sequence (4) in F. (iii) If (4) is a d-auslander Reiten sequence in F, then f d+1 = τ d (f 0 ) and f 0 = τ d (f d+1). 20

7 (d + 2)-angulated categories Definition 7.1 (Geiss Keller Oppermann [2, def. 2.1]). Let C be an additive category with an automorphism Σ d. The inverse is denoted Σ d, but Σ d is not assumed to be a power of another functor. A Σ d -sequence in C is a diagram of the form c 0 γ 0 c 1 c 2 c d d+1 γd+1 c Σ d (c 0 ). (5) Definition 7.2 (Geiss Keller Oppermann [2, def. 2.1]). A (d + 2)- angulated category is a triple (C, Σ d, ) where is a class of Σ d - sequences, called (d + 2)-angles, satisfying the following conditions. (N1) is closed under sums and summands and contains c id c c 0 0 0 Σ d (c) for each c C. For each morphism c 0 γ 0 c 1 in C, the class contains a Σ d -sequence of the form (5). (N2) The Σ d -sequence (5) is in if and only if so is its left rotation c 1 c 2 d+1 γd+1 c Σ d (c 0 ) ( 1)d Σ d (γ 0 ) Σ d (c 1 ). 21

(N3) A commutative diagram with rows in has the following extension property. b 0 β 0 b 1 b 2 b d 1 b d b d+1 Σ d (b 0 ) c 0 c 1 c 2 c d 1 c d c d+1 Σ d (c 0 ) (N4) The octahedral axiom (the following version of the axiom is equivalent to the one in [2] by [7, thm. 2.2]): A commutative diagram with rows in has the following extension property, ξ x ξ y ζ x υ a 2 z b 2 a 3 b 3 a d b d a d+1 b d+1 Σ d (x) Σ d (x) Σ d (ξ) y υ z c 2 c 3 c d c d+1 δ Σ d (y), where each upright square is commutative and the totalisation a 2 a 3 b 2 a 4 b 3 c 2 a d+1 b d c d 1 b d+1 c d c d+1 Σd (ζ)δ Σ d (a 2 ) Σ d (β 0 ) is a (d + 2)-angle. 22

The following is a sample of what can be concluded from the definition. Proposition 7.3 (Geiss Keller Oppermann [2, prop. 2.5(a)]). Let (C, Σ d, ) be a (d+2)-angulated category, c C an object. Each (d + 2)-angle (5) induces long exact sequences C ( c, Σ d (c d+1 ) ) C (c, c 0 ) C (c, c d+1 ) C ( c, Σ d (c 0 ) ) and C ( Σ d (c 0 ), c ) C (c d+1, c) C (c 0, c) C ( Σ d (c d+1 ), c ). We would like to be able to obtain (d + 2)-angulated categories from d-abelian categories: Question 7.4 (The higher derived category problem). Let F be a suitable d-abelian category. Does there exist a d-derived category of F, in the sense that there is a (d + 2)-angulated category C with the following properties? (i) C contains F as an additive subcategory. (ii) Each d-exact sequence (3) in F induces a (d + 2)-angle (5) in C. (iii) We have Ext di F(f, f ) = C ( f, (Σ d ) i (f ) ) where i 0 is an integer and f, f F. The Ext group is defined by means of Yoneda extensions. 23

(iv) Each d-exact sequence in F induces two long exact sequences of Ext-groups (note that this is only true for suitable F ). Each (d + 2)-angle in C induces the long exact sequences in Proposition 7.3. The long exact sequences are identified under the isomorphisms in (iii). However, the answer to Question 7.4 is unknown. Here is a method we do have for obtaining (d + 2)-angulated categories: Theorem 7.5 (Geiss Keller Oppermann [2, thm. 1]). Let D be a triangulated category with suspension functor Σ. Let C be a d-cluster tilting subcategory of D which satisfies Σ d (C ) C. Then (C, Σ d, ) is a (d+2)-angulated category, where consists of all Σ d -sequences (5) which can be obtained from diagrams of the form c 1 c 2 c d 1 c d c 0 x 1.5 x 2.5 A wavy arrow x x d 0.5 c d+1. y signifies a morphism x Σ(y), and the d+1 γd+1 composition of the wavy arrows is c Σ d (c 0 ) in (5). Each oriented triangle is a triangle in D, and each non-oriented triangle is commutative. 24

This permits answering Question 7.4 in a special case: Theorem 7.6 (Iyama). Let Φ be a d-representation finite algebra, and let F be the unique d-cluster tilting subcategory of mod(φ), cf. Theorem 4.8. Then F has a d-derived category C in the sense of Question 7.4. Remark 7.7. What Iyama in fact proved is that the subcategory C = add{ Σ dj f j Z, f F } D b (mod Φ) is d-cluster tilting. It clearly satisfies Σ d (C ) C, so is (d + 2)- angulated by Theorem 7.5. It is also clear that C has property (i) in Question 7.4, and it can be proved that it satisfies (ii) (iv) as well. Exercise 7.8. Prove Proposition 7.3. 25

A A class of examples by Vaso All the results in this appendix are due to Vaso, see [8, sec. 4]. Definition A.1. Let d 2, l 2, m 3 be integers such that m 1 l = d 2 and such that d is even if l > 2. Let Q be the quiver m m 1 2 1. In this section we set Φ = CQ/(rad CQ) l. Equivalently, Φ is the path algebra of the quiver Q with the relations that l consecutive arrows compose to zero. Remark A.2. The indecomposable projective and injective modu- 26

les in mod(φ), shown as representations of Q, are the following. f 1 = 0 0 0 0 0 0 0 0 0 0 0 C f 2 = 0 0 0 0 0 0 0 0 0 0 C C. projective f l 1 = 0 0 0 0 0 0 0 0 0 C C }{{} l 1 f l = 0 0 0 0 0 0 0 0 C C C }{{} f l+1 = 0 0 0 0 0 0 0 C C C 0 }{{} l. f m 1 = 0 C C C 0 0 0 0 0 0 0 }{{} l f m = C C C 0 0 0 0 0 0 0 0 }{{} l f m+1 = C C 0 0 0 0 0 0 0 0 0 }{{} l 1. f m+l 2 = C C 0 0 0 0 0 0 0 0 0 0 f m+l 1 = C 0 0 0 0 0 0 0 0 0 0 0 Among them, the indecomposable projective modules are l projectiveinjective injective f i = e i Φ for 1 i m where e i is the idempotent corresponding to the vertex i in Q. The 27

f i appear in the Auslander Reiten quiver of mod(φ) as follows. f 1 f 2 f l 1 f l f l+1......... f m 1 f m f m+1......... f m+l 2 f m+l 1 It will be convenient to set f i = 0 if i 0 or i m + l. A.i The d-abelian category of Φ The following result shows that there is a canonical d-abelian category F associated to Φ, cf. Theorem 4.8. Theorem A.3. The algebra Φ is d-representation finite. The unique d-cluster tilting subcategory of mod(φ) is F = add(φ DΦ) = add { f i 1 i m + l 1 }. Example A.4. d = 4, l = 4, m = 9. f 1 f 2 f 3 f 4 a f 5 f 6 b f 7 f 8 c f 9 f 10 f 11 f 12 28

There is a short exact sequence and similarly 0 f 1 = 0 0 0 0 0 0 0 0 C f 4 = 0 0 0 0 0 C C C C a = 0 0 0 0 0 C C C 0 0, 0 a f 5 b 0, 0 b f 8 c 0, 0 c f 9 f 12 0. The short exact sequences can be spliced to give augmented F - resolutions of a, b, and c in the sense of Definition 3.5. 0 f 1 f 4 a 0, 0 f 1 f 4 f 5 b 0, 0 f 1 f 4 f 5 f 8 c 0. The short exact sequences can also be spliced to an exact sequence 0 f 1 f 4 f 5 f 8 f 9 f 12 0 which is a 4-exact sequence in F, see Definition 5.1. In particular, several 4-kernels and 4-cokernels can be read off. Observe that a 29

4-kernel does not necessarily have 4 non-zero objects; for instance, 0 0 f 1 f 4 f 5 is a 4-kernel of f 5 f 8. Proposition A.5. (i) We have dim C Hom Φ (f i, f j ) = { 1 if 0 j i l 1, 0 otherwise, (ii) If i q j then each morphism f i f j factors through f q. (iii) The quiver of F is f 1 f 2 f 3 f m+l 3 f m+l 2 f m+l 1 where the composition of l consecutive arrows is zero. (iv) If 0 j i l 1 then each non-zero morphism f i f j fits into an exact sequence f j l f i f j f i+l. Remark A.6. Part (iii) is immediate from parts (i) and (ii). Part (iv) implies that if j l 1, then f i f j is an injection because f j l = 0. Similarly, if i m then f i f j is a surjection because f i+l = 0. 30

Proposition A.7. If 0 j i l 1 then there is an exact sequence f i l f j l f i f j f i+l f j+l. Proof. Iterate Proposition A.5(iv). Remark A.8. We can follow the sequence in Proposition A.7 left or right until we reach a zero module. This gives a d-kernel or a d-cokernel of f i f j. Proposition A.9. We have τ d (f j ) = { 0 if 1 j m, f j m if m + 1 j m + l 1. Proof. If 1 j m then f j is projective whence τ d (f j ) = 0. Suppose m + 1 j m + l 1. There is a surjection f m f j which fits into an exact sequence from Proposition A.7: f m 2l f j 2l }{{} 2 f m l f j l }{{} 1 f m f j }{{} 0 0. The numbered brackets are useful for bookkeeping. Since m 1 l = d 2 31

we have m d 2l = 1, so on the left the sequence contains which can also be written 0 f 1 f j d 2 l }{{} d 2 0 f 1 f }{{ j+1 m }. (6) d 2 The exact sequence is an augmented projective resolution of f j because each term except f j is projective. The non-zero terms in Equation (6) are p d p d 1. We can write them e 1 Φ e j+1 m Φ. Applying ( ) and augmenting with the cokernel gives an exact sequence Φe j+1 m Φe 1 Tr d (f j ) 0. These left modules can be viewed as representations of the quiver Q op = m m 1 2 1. Then Φe j+1 m is supported on the vertices j + 1 m through m while Φe 1 is supported on the vertices 1 through m. Hence Tr d (f j ) is 32

supported on the vertices 1 through j m. This finally means that τ d (f j ) = DTr d (f j ), viewed as a representation of Q, is supported on the same vertices, 1 through j m, and hence τ d (f j ) = e j m Φ = f j m. Proposition A.10. If m + 1 j m + l 1 then the d- Auslander Reiten sequence ending in f j is 0 f j m f j+1 m f j 1 l f j l f j 1 f j 0. A.ii The (d + 2)-angulated category of Φ Combining Remark 7.7 and Theorem A.3 shows that C = add{ Σ dj f j Z, f F } D b (mod Φ) is a (d + 2)-angulated category which can be viewed as the d-derived category of the d-abelian category F from Subsection A.i. Proposition A.11. (i) Up to isomorphism, the indecomposable objects of C are the Σ dj f i where i, j are integers with 1 i m + l 1. 33

(ii) The quiver of C is Σ d f 1 Σ d f m+l 1 f 1 f m+l 1 Σ d f 1 Σ d f m+l 1 where the composition of l consecutive arrows is zero. Exercise A.12. Prove Proposition A.5. Exercise A.13. Show that the sequence in Proposition A.10 is indeed a d-auslander Reiten sequence. Exercise A.14. Prove Proposition A.11. References [1] E. E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981), 189 209. [2] C. Geiss, B. Keller, and S. Oppermann, n-angulated categories, J. Reine Angew. Math. 675 (2013), 101 120. [3] O. Iyama, Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories, Adv. Math. 210 (2007), 22 50. [4] O. Iyama and S. Oppermann, n-representation-finite algebras and n-apr tilting, Trans. Amer. Math. Soc. 363 (2011), 6575-6614. 34

[5] G. Jasso, n-abelian and n-exact categories, Math. Z. 283 (2016), 703 759. [6] G. Jasso and S. Kvamme, An introduction to higher Auslander Reiten theory, preprint (2016). math.rt/1610.05458v1 [7] Z. Lin and Y. Zheng, Homotopy cartesian diagrams in n- angulated categories, preprint (2016). math.rt/1612.08903v1 [8] L. Vaso, n-cluster tilting subcategories of representation-directed algebras, preprint (2017). math.rt/1705.01031v1 35